2 Answers
2

It depends on the words and how they are pronounced, although the pattern you noticed is common.

For example, the following don't fit the pattern you see:

Brad Pitt -> ブラピ

Ice Cream -> アイス

Convenience Store -> コンビニ

There has been a lot of linguistic studies about this and many patterns that exist.
One rule is that they are always truncated to 2 to 4 mora. Also, [濁音]{だくおん} are avoided if possible and generally heavy syllables are avoided at the end.

Another example of the first I came across is ハピバ（Happy Birthday). BTW: That looks like quite a paper but, if you forgive me asking before I read it, what is a mora, a dakuon (voiced consonant?) and a heavy syllable? (I am sure the answers are easy but I have learnt on this website that I am not a proper linguist!)
– TimSep 7 '12 at 23:01

@Tim: Those are all linguistic terms and have a corresponding wikipedia article. I've never properly studied linguistics myself, so the reading the wikipedia article or googling it would probably give a better explanation than I can provide.
– Jesse GoodSep 7 '12 at 23:09

In case anybody has the same qs as me:[but BTW if this is not clear then I recommend looking it up as Jesse Good suggests because there is a bit more to it than I've said but it is enough for me on this subject.]: ド has 1 mora and is 1 syllable, ドン is also 1 syllable but has 2 mora ("it has a longer sound"). A voiced consonant (vs an unvoiced one) is the difference between the sound of the letters "ssss" and the letter "zzz". (If you touch your Adam's apple you can feel the vibration when you make the "z" sound: this is the sound of one voiced consonant, other examples include "l", "m" & "n".)
– TimSep 8 '12 at 3:47

It is common to abbreviate a loanword to four morae, and when the original loanword is a compound word consisting of two components, then it is common to do so by taking the first two morae from each component. Sometimes sokuon (little tsu: ッ) and chōon (ー) are skipped when counting two morae for this purpose.

リモートコントロール (remote control) or リモートコントローラー (remote controller) → リモコン

パーソナルコンピューター (personal computer) → パソコン

Sometimes the final sokuon or chōon after abbreviating is dropped, and this explains ブラピ in Jesse Good’s answer and ハピバ in Tim’s comment on Jesse Good’s answer.

But not every loanword is abbreviated like this, as Jesse Good already wrote.

This pattern is not really specific to loanwords. I have seen this pattern applied to the title of comics, video games, and TV programs. In some sense, even abbreviations such as [外国]{がいこく}[為替]{かわせ} to [外為]{がいため} may be viewed as an instance of the same pattern.

By the way, I do not think that ドンマイ is an abbreviation of something like ドントマインド. It seems to me more like a transcription of the pronunciation of the phrase “Don’t mind” heard by Japanese speakers.